Compressive sampling, also known as compressed sampling, compressed sensing or compressive sensing, is a data sampling technique which often exhibits improved efficiency relative to conventional Nyquist sampling. Compressive sampling may be characterized mathematically as multiplying an N-dimensional data vector x by the conjugate-transpose of an N×M dimensional sampling matrix Ψ to yield an M-dimensional compressed measurement vector y, where typically M is much smaller than N. If the data vector x happens to be sparse in some domain that is linearly related to x, then x can be recovered from the compressed measurement vector y provided that the sampling matrix Ψ satisfies the so-called restricted isometry property. For additional details see, for example, E. J. Candès and M. B. Wakin, “An Introduction to Compressive Sampling,” IEEE Signal Processing Magazine, Vol. 25, No. 2, March 2008, and E. J. Candès, “Compressive Sampling,” Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006, both of which are incorporated by reference herein. As sparse signals constitute a large majority of signals encountered in nature as well as in man-made systems, compressive sampling is expected to come into increasingly widespread use for efficient transmission and storage of such signals.
Although it is known that sparse signals can be efficiently transmitted, stored or otherwise processed using compressive sampling, such processing does not necessarily provide adequate levels of security. Security in the context of compressive sampling is addressed, by way of example, in A. Orsdemir et al., “On the Security and Robustness of Encryption Via Compressed Sampling,” IEEE Military Communications Conference (MILCOM) 2008, San Diego, Calif., Nov. 16-19, 2008, pp. 1-7, J. Wen, “Key Issues in Secure, Error Resilient Compressive Sensing of Multimedia Content,” IEEE International Conference on Multimedia and Expo (ICME) 2009, New York, N.Y., Jun. 28-Jul. 3, 2009, pp. 1590-1591, and Y. Rachlin and D. Baron, “The Secrecy of Compressed Sensing Measurements,” 46th Annual Allerton Conference on Communication, Control, and Computing, Urbana-Champaign, Ill., Sep. 23-26, 2008, pp. 813-817, all of which are incorporated by reference herein. However, conventional compressive sampling as described in the above-cited references fails to provide sufficient security for transmission, storage or other processing of the compressed measurement vector y, such that the data vector x can be recovered only by authorized parties, while maintaining the overall efficiency of the compressive sampling process.